Interests
My interests are in probability,
functional analysis and statistics. At the moment, most of my research centers
on numerical analysis, stochastic processes and simulation.
Books
If you are interested in analysis and probability with a view to mathematical economics then here is my reading list. I am extremely fussy, and I think that all of these books are great.
Introductory
- Introduction to Real Analysis, Bartle and Sherbet
- Abstract Analysis, Will Light
Intermediate and Graduate texts
- An Introduction to Measure and Probability, J. C. Taylor
- Introductory Real Analysis, Kolmogorov and Fomin
- Probability with Martingales, D. Williams
- A Users Guide to Measure-Theoretic Probability, Pollard
- Principles of Real Analysis, Aliprantis and Burkinshaw
- Real Analysis and Probability, R. M. Dudley
- Measures, Integrals and Martingales by R. L. Schilling
- Analysis for Applied Mathematics, W. Cheney
Markov Chains
- Chaos, Fractals and Noise: Stochastic Aspects of Dynmaics, Lasota and Mackey
- Markov Chains and Stochastic Stability, Meyn and Tweedie.
- Discrete-Time Markov Control Processes, Hernandez-Lerma and Lassere
Monographs, Other Topics
- Optimization by Vector Space Methods, Luenberger.
- Infinite Dimensional Analysis, Aliprantis and Border
- Lectures on the Coupling Method, Lindvall
On-line Resources
- MIT Lectures on Probability Theory
- Collection of free maths books: Great collection of free maths books as PDF files
The book by Bartle and Sherbert is where you should start. It is the best introduction to analysis that I know. Kolmogorov and Fomin is a nice read on functional analysis. Pollard has a great style and interesting topics. He uses a non-standard notation that takes a while to get your head around, but it's been kind of revolutionary for me in a way. Aliprantis and Burkinshaw is an excellent graduate text with lots of exercises. Dudley is great, but he frightens me a little bit. How can anyone be that smart? Schilling is both beautiful and useful. Cheney is a nice balance between theory and applications.
Meyn and Tweedie's book is available on-line. With apologies to any followers of the major monotheistic religions, I do believe that this is the single GREATEST BOOK in the history of the world. Sean Meyn is one of the top few researchers on stability of Markov chains. Sadly, Richard Tweedie recently passed away, but he made a huge contribution to Markov chain theory.
For background reading, try the wonderful book called "In Search of
Infinity" by N. Ya. Vilenkin.
Other Bits and Pieces
I love the work of Escher, especially this one. Here's an interesting little equation about girls and evil.
The man generally recognized as the founder of modern functional analysis is
Stefan
Banach. He is very smart because he is POLISH. Many definitions and
results in normed linear space stem from his 1922 thesis. Another important
contributor to the theory of abstract spaces was Maurice
Frechet. Frechet defined metric space for the first time in his thesis.
He also helped generalize the Lebesgue integral to operate on functions of
arbitrary domain.